A070048 -id:A070048 - cp's OEIS frontend

A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 3, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 4, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 5, 3, 0

Offset: 0

Ilya Gutkovskiy, Jul 26 2017

A(n,k) is the number of partitions of n into distinct parts where no part is a multiple of k.

			Square array begins:
  1,  1,  1,  1,  1,  1, ...
  0,  1,  1,  1,  1,  1, ...
  0,  0,  1,  1,  1,  1, ...
  0,  1,  1,  2,  2,  2, ...
  0,  1,  1,  1,  2,  2, ...
  0,  1,  2,  2,  2,  3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=1-10 give: A000007, A000700, A003105, A070048, A096938, A261770, A097793, A261771, A112193, A261772.

Cf. A286653.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)/(1 + x^(i k)), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x]/QPochhammer[-1, x^k], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

For asymptotics of column k see comment from Vaclav Kotesovec in A261772.

A301507 Expansion of Product_{k>=0} (1 + x^(4k+1))(1 + x^(4*k+2)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 13, 14, 14, 16, 18, 20, 23, 24, 27, 30, 31, 34, 37, 41, 46, 49, 53, 58, 62, 67, 73, 80, 88, 94, 101, 109, 117, 127, 136, 147, 161, 172, 184, 198, 211, 228, 245, 262, 284, 304, 324, 347, 370, 397, 425, 454, 488

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 1 or 2 mod 4.

			a(11) = 3 because we have [10, 1], [9, 2] and [6, 5].

Index entries for sequences related to partitions

Cf. A035365, A042963, A070048, A108932, A169975, A301504, A301505, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k + 1)) (1 + x^(4 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x, x^4] QPochhammer[-x^2, x^4], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 2}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042963(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A301508 Expansion of Product_{k>=0} (1 + x^(4k+2))(1 + x^(4*k+3)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 4, 4, 5, 5, 6, 7, 6, 8, 9, 9, 11, 12, 13, 14, 15, 17, 19, 20, 23, 25, 27, 29, 31, 35, 37, 40, 46, 48, 52, 57, 60, 66, 71, 76, 85, 90, 97, 105, 112, 121, 129, 140, 152, 161, 174, 187, 198, 214, 228, 245, 265, 280, 302, 323, 342

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 2 or 3 mod 4.

			a(13) = 3 because we have [11, 2], [10, 3] and [7, 6].

Index entries for sequences related to partitions

Cf. A035366, A042964, A070048, A131795, A147599, A301504, A301505, A301507.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k + 2)) (1 + x^(4 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x^2, x^4] QPochhammer[-x^3, x^4], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{2, 3}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042964(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A304628 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(4*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 47, 181, 729, 2948, 12031, 49540, 205153, 853546, 3565505, 14943839, 62810786, 264650683, 1117486463, 4727486583, 20032950744, 85017558081, 361289789377, 1537198394570, 6547611493822, 27917246924099, 119141276756545, 508884954441331, 2175284934712217, 9305217981192748

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A070048, A255526, A285290, A296044, A296164, A304629.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 + x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[Product[((1 - x^(8 k - 4))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
(* Calculation of constants {d,c}: *) With[{k = 4}, {1/r, Sqrt[QPochhammer[-1, (r*s)^k] / (2*Pi*(r^2*s*Derivative[0, 2][QPochhammer][-1, r*s] - k^2*(r*s)^(2*k) * Derivative[0, 2][QPochhammer][-1, (r*s)^k] - k*(1 + k)*(r*s)^k * Derivative[0, 1][QPochhammer][-1, (r*s)^k]))]} /. FindRoot[{s == QPochhammer[-1, r*s]/QPochhammer[-1, (r*s)^k], QPochhammer[-1, (r*s)^k] + k*(r*s)^k*Derivative[0, 1][QPochhammer][-1, (r*s)^k] == r*Derivative[0, 1][QPochhammer][-1, r*s]}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 - x^(8*k-4))/(1 - x^(2*k-1)))^n.

a(n) ~ c * d^n / sqrt(n), where d = 4.3582188263213968630940316689... and c = 0.266443662680498334500839... - Vaclav Kotesovec, May 18 2018

A332310 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 4.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 9, 5, 12, 17, 23, 43, 50, 55, 67, 111, 144, 273, 291, 377, 410, 689, 827, 961, 1860, 1663, 2647, 3573, 4610, 4683, 6753, 8465, 11232, 16835, 19985, 24073, 29258, 40411, 51367, 58431, 72084, 99807, 119409, 176387, 199922, 250841, 290123

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Index entries for sequences related to compositions

Cf. A001935, A032020, A032021, A070048, A332309, A332311.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 4], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A116498 Expansion of psi(-q)/psi(-q^2) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -2, 1, -2, 3, -3, 4, -5, 6, -7, 8, -9, 11, -13, 16, -18, 21, -24, 27, -32, 36, -41, 48, -54, 61, -70, 78, -88, 100, -112, 127, -143, 159, -179, 199, -222, 248, -276, 308, -342, 380, -421, 465, -516, 570, -629, 697, -767, 845, -932, 1022, -1124, 1236, -1355, 1488, -1631, 1785, -1954, 2136

Offset: 0

Michael Somos, Feb 18 2006

sign

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000122, A000700, A010054, A070048, A121373.

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[ q^(1/8)* eta[q]*eta[q^4]^2/(eta[q^2]^2*eta[q^8]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 07 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)^2/eta(x^2+A)^2/eta(x^8+A), n))}

Expansion of q^(1/8)*eta(q)*eta(q^4)^2/(eta(q^2)^2*eta(q^8)) in powers of q.
a(n)=(-1)^n*A070048(n).
Euler transform of period 8 sequence [ -1,1,-1,-1,-1,1,-1,0,...].
Given g.f. A(x), then B(x)=A(x^8)/x satisfies 0=f(B(x),B(x^3)) where f(u,v)=3*u*v -(u+v^3)*(v-u^3).
G.f.: Product_{k>0} (1+x^(2k))/((1+x^k)(1+x^(4k))) = (Sum_{k>0} (-x)^((k^2-k)/2))/(Sum_{k>0} (-x^2)^((k^2-k)/2)).

cp's OEIS Frontend

A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A301507 Expansion of Product_{k>=0} (1 + x^(4*k+1))*(1 + x^(4*k+2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A301508 Expansion of Product_{k>=0} (1 + x^(4*k+2))*(1 + x^(4*k+3)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A304628 a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 + x^(4*k)))^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A332310 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A116498 Expansion of psi(-q)/psi(-q^2) in powers of q where psi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A301507 Expansion of Product_{k>=0} (1 + x^(4k+1))(1 + x^(4*k+2)).

A301508 Expansion of Product_{k>=0} (1 + x^(4k+2))(1 + x^(4*k+3)).